Understanding Exponents, Divisibility, and Fractions
Understanding Exponents
Exponentiation is a multiplication of equal factors. For example: 2.2.2.2.2 = 25 (where 5 is the exponent and 2 is the base). The base indicates how many times it is multiplied by itself.
- 43 = 4.4.4 = 64
- 8.8.8.8.8 = 85
- Base 2, exponent 6 = 64
- Base 0, exponent 9 = 0
Order of operations with exponents:
- Powers
- Multiplications
- Additions and subtractions
For example: 3.24 + 25 = 3.16 (= 24) + 32 (= 25) = 48 + 32 = 80.
Properties of Exponentiation
- Multiplication of powers with the same base:
Statistics and ggplot2: Quick Reference
Statistics and ggplot2: Quick Reference
Central Tendency
- Mean: The average of values, affected by outliers.
- Formula: \(\bar{x} = \frac{\Sigma x_i}{n}\)
- Median: The middle value, robust to outliers.
- Mode: The most frequent value in a dataset.
Variability Metrics
- Range: \(\text{Max} – \text{Min}\)
- Population Variance: \(\sigma^2 = \frac{\Sigma (x_i – \mu)^2}{N}\)
- Sample Variance: \(s^2 = \frac{\Sigma (x_i – \bar{x})^2}{n-1}\) (Bessel’s correction).
- Standard Deviation (SD): The square root of variance.
- Formula:
Statistical Inference: Z-Distribution, T-Distribution, and Regression
Chapter 6: Standard Error (SE)
The standard error (SE) is the standard deviation of the sampling distribution of a statistic. It measures the precision of the sample statistic as an estimate of the population parameter. A z-distribution is the standard normal distribution with a mean of 0 and a standard deviation of 1. It is used for testing hypotheses about a single population mean or proportions when σ is known. The T-distribution is a family of distributions that are similar to the normal distribution
Read MoreVehicle Tax Analysis: Price, Age, and Regression Insights
1. Interpreting the Slope in the Simple Linear Regression Model
A 1% increase in price is associated with a 0.8% increase in taxes. Given that the increase is less than 1%, the vehicle tax is regressive, not progressive, meaning that more expensive cars pay proportionally less tax.
rate = exp(b1) * exp(0.8161 * log_price) = exp(b1) * (exp(log_price))^0.8161 = exp(b1) * (price)^0.8161
Hence, an increase in the price of 1% implies an increase in the rate of (1.01)^0.8161 = 1.00815, that is, an increase
Read MoreMastering Math: Exponents, Roots, Polynomials, and More
Powers
Power = (Base)exponent
Properties of Powers
- 1st
- 2nd
- 3rd
- 4th
- 5th
- 6th
- 7th
- 8th
- 9th
Scientific Notation
A) Mt indicates the number of zeros to the right.
B) Mor indicates tenths. No. If I have a high ten, and M is positive, add zeros to the number as indicated by the M.
Roots
A) Numeric values of a radical. If the radical is a positive number, the solution is a unique positive root.
B) If the radical is negative and the index is even, the solution is a negative root.
C) Based on a positive, even index, there is
Read MoreHypothesis Theory: Types, Characteristics, and Formulation
Hypothesis Theory
A hypothesis is an assumption that establishes a relationship between two or more variables, expressed as facts, events, or factors. It must be tested to be accepted as valid.
Role of a Hypothesis
- Guides and directs an investigation.
- Assumptions should be deduced from the problem and aims of the study.
- Must be consistent with the theoretical framework.
- Determines the type of study and design methodology.
Characteristics of a Hypothesis
- Must refer to a real situation with a defined context.